If I roll 5 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number?
Solution: While it is almost tempting to make this into the casework problem to end all casework problems, using complementary probability makes the problem a lot simpler, so we will find the probability that the product is not composite.

It's clear that if more than one die shows a roll greater than 1, the resulting product will have more than one factor that is greater than 1 and therefore be composite. Additionally, if any of the dice show a 4 or a 6, the product will obviously be composite. So in order for the product to not be composite, we need at least four dice to be a 1 and the fifth die to either be another 1 or a 2, a 3, or a 5.

There's exactly one way in which all the the dice can be 1's, and for each of the other three cases there are 5 ways since the die that isn't a 1 can appear in any of the 5 other positions, so there are a total of $3\cdot5+1=16$ ways of rolling the dice that result in a product that isn't composite. Additionally, there are 6 options for the roll of each of the 5 dice for a total of $6^5=7776$ possible outcomes, which means that the probability of the product not being composite is $\dfrac{16}{7776}$ and the probability that it is composite is $1-\frac{16}{7776}=\frac{7760}{7776}=\boxed{\frac{485}{486}}$.

Note: Many students have tried to argue that 1 is not a prime number and should not be counted. As we are doing complementary probability, it is important to consider all numbers that are not composite, and 1 is definitely not composite.